We develop a systematic study of the schur tensor product both in the category of operator spaces and in that of C * -algebras.

For C * -algebras A and B, we study the bi-continuity of the canonical embedding of A * * ⊗γ B * * (A * * ⊗B * * ) into (A ⊗γ B) * * (resp. (A ⊗B) * * ), and its isomorphism. Ideal structure of A ⊗B has been obtained in case A or B has only finitely many closed ideals.2010 Mathematics Subject Classification. Primary 46L06, Secondary 46L07,47L25. Key words and phrases. Banach space projective tensor norm, Operator space projective tensor norm.

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Abstract. We extend the λ-theory of operator spaces given in [4], that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach * -algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to λ for the algebraic tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of λ-tensor product of C * -algebras has also been discussed.

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product A ⊗B (respectively the Haagerup tensor product A ⊗ h B) to be Arens regular are obtained. Using the non-commutative Grothendieck's inequality, we show that, for C * -algebras A and B, the Arens regularity of Banach algebras A ⊗ h B, A ⊗ γ B, A ⊗ s B and A ⊗B are equivalent, where ⊗ h , ⊗ γ , ⊗ s and ⊗ are the Haagerup, the Banach space projective tensor norm, the Schur tensor norm and the operator space projective tensor norm, respectively.